Asymptotic normality of variance estimator in a heteroscedastic model with dependent errors

Abstract

Consider the heteroscedastic regression model Y ni =g(x ni )+σ ni ε ni (1≤i≤n), where , the design points (x ni , u ni ) are known and nonrandom, g(·) and f(·) are unknown functions defined on [0, 1], and the random errors {ε ni , 1≤i≤n} are assumed to have the same distribution as {ξ i , 1≤i≤n}, which is a stationary and α-mixing time series with Eξ i =0. Under appropriate conditions, we study the asymptotic normality of an estimator of the function f(·). At the same time, we derive a Berry–Esseen-type bound for the estimator. As a corollary, by making a certain choice of the weights, the Berry–Esseen-type bound of the estimator can attain O(n −1/12(log n)−1/3). Finite sample behaviour of this estimator is investigated too.